Aguero was carried off on a stretcher after eight minutes following a challenge by Saints’ Nathaniel Clyne during Sunday’s 3-2 win over Southampton.
“I don’t think Sergio will be out for long and we are hoping he will be back after the international break,” Mancini told the Man City website.
“It is obviously fantastic news for him and for us. We were very worried a couple of days ago because we feared he could be out for six months or more.”
He added: “That would have been very bad news.
“It looked like a very strange injury at the time and because it was his knee, we were obviously very concerned so this news is a relief for everybody.”
City get break in that their is an international break between 2 and 15 September, meaning could miss as few as two Premier League matches;Sunday’s clash at Liverpool and then the visit to QPR six days later.
Also expected back after the international break are Gareth Barry and Micah Richards. Barry suffered an abdominal injury that ruled him out of Euro 2012, but is now back in training and working to catch up after missing the pre-season, while Richards is making progress after suffering an ankle injury while playing for Team GB at the Olympics.
“Gareth will also be ready after the international break,” Mancini said.
“He hasn’t played any games for almost four months and he needs to build his fitness up, but he is training again and I don’t see him being out for any longer than one month and hopefully, maybe just a couple of weeks.
“Micah’s injury is more serious and he could be out a little longer, but having said that, with rest and the right treatment, we are hopeful he could be ready in about one month, which again would be great news.”